Presenter: Yufan Liu yliu33@kent.edu November 17th, 2011 1

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An Introduction to the Kalman Filter

Presenter: Yufan Liuyliu33@kent.eduNovember 17th, 2011

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OutlineBackgroundDefinitionApplicationsProcessesExampleConclusion

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Low and high pass filtersLow pass filter allows passing low frequency

signals. It can be used to filter out the gravity. High pass filter allows high frequency signals

to pass. Band pass filter is a combination of these

two.They are working on frequency domain.

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Definition of Kalman filterIt is an optimal (linear) estimator or

optimal recursive data processing algorithm.

Belongs to the state space model(time domain) compared to frequency domain

Components: system's dynamics model, control inputs, and recursive measurements(include noise)

Parameters include indirect, inaccurate and uncertain observations.

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Typical Kalman filter application

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Applications

http://en.wikipedia.org/wiki/Apollo_program

http://www.lorisbazzani.info/research-2/

http://en.wikipedia.org/wiki/Gps

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Hidden Markov ModelMarkov Property :The next n+1 depends on n

but not the entire past(1…n-1) The state is not clearly visible, but the output

is visibleThe states give us information on the system. The task is to derive the maximum likelihood

of the parameters

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HMM example

http://en.wikipedia.org/wiki/Hidden_Markov_model

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Major equation

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Step 1: Build a model

Any xk is a linear combination of its previous value plus a control signal uk and a process noise.

The entities A, B and H are in general matrices related to the states. In many cases, we can assume they are numeric value and constant.

Wk-1 is the process noise and vk is the measurement noise, both are considered to be Gaussian.

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Step 2: Start process

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Step 3: Iterate

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A simple exampleEstimate a random constant:” voltage”

reading from a source.It has a constant value of aV (volts), so there

is no control signal uk. Standard deviation of the measurement noise is 0.1 V.

It is a 1 dimensional signal problem: A and H are constant 1.

Assume the error covariance P0 is initially 1 and initial state X0 is 0.

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A simple example – Part 1

Time 1 2 3 4 5 6 7 8 9 10

Value 0.39

0.50 0.48 0.29 0.25 0.32 0.34 0.48 0.41 0.45

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A simple example – Part 2

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A simple example – Part 3

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Result of the example

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The Extended Kalman filter In simple cases, such as the linear dynamical

system just, exact inference is tractable; however, in general, exact inference is infeasible, and approximate methods must be used, such as the extended Kalman filter.

Unlike its linear counterpart, the extended Kalman filter in general is not an optimal estimator

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Properties and conclusionIf all noise is Gaussian, the Kalman filter

minimizes the mean square error of the estimated parameters

Convenient for online real time processing.Easy to formulate and implement given a

basic understanding.To enable the convergence in fewer steps:

Model the system more elegantlyEstimate the noise more precisely

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ReferenceLindsay Kleeman, Understanding and

Applying Kalman Filtering, Department of Electrical and Computer Systems Engineering, Monash University, Clayton

Peter Maybeck, Stochastic Models, Estimation, and Control, Volume 1

Greg Welch, Gary Bishop, "An Introduction to the Kalman Filter",  University of North Carolina at Chapel Hill Department of Computer Science, 2001

Thank you